Abstract

This thesis presents resource-bounded category and resource bounded
measure - two new tools for computational complexity theory - and
some applications of these tools to the structure theory of exponential
complexity classes.

Resource-bounded category, a complexity-theoretic version of the
classical Baire category method, identifies certain subsets of PSPACE, E,
ESPACE, and other complexity classes as meager. These meager sets are
shown to form a nontrivial ideal of "small" subsets of the complexity class.
The meager sets are also (almost) characterized in terms of certain two-person
infinite games called resource-bounded Banach-Mazur games.

Similarly, resource-bounded measure, a complexity-theoretic version of
Lebesgue measure theory, identifies the measure 0 subsets of E, ESPACE,
and other complexity classes, and these too are shown to form nontrivial
ideals of "small" subsets. A resource-bounded extension of the classical
Kolmogorov zero-one law is also proven. This shows that measurable sets of
complexity-theoretic interest either have measure 0 or are the complements of
sets of measure 0.

Resource-bounded category and measure are then applied to the
investigation of uniform versus nonuniform complexity. In particular,
Kannan's theorem that ESPACE P/Poly is extended by showing that P/ Poly
∩ ESPACE is only a meager, measure 0 subset of ESPACE. A theorem of
Huynh is extended similarly by showing that all but a meager, measure 0
subset of the languages in ESPACE have high space-bounded Kolmogorov
complexity.

These tools are also combined with a new hierarchy of exponential time
complexity classes to refine known relationships between nonuniform
complexity and time complexity.

In the last part of the thesis, known properties of hard languages are
extended. In particular, recent results of Schoning and Huynh state that any
language L which is ≤^P_m -hard for E or ≤^P_T -hard for ESPACE cannot be
feasibly approximated (i.e., its symmetric difference with any feasible language
has exponential density). It is proven here that this conclusion in fact holds
unless only a meager subset of E is ≤^P_m -reducible to L and only a meager,
measure 0 subset of ESPACE is ≤^(PSPACE)_m -reducible to L. (It is conjectured,
but not proven, that this result is actually stronger than those of Schoning
and Huynh.) This suggests a new lower bound method which may be useful in
interesting cases.